An Integration Algorithm For General Isotropic Elastoplasticity In Invariant Space

This thesis develops an efficient return mapping algorithm for implicit integration of general isotropic elastoplastic constitutive equations in invariant space.Two sets of three mutually orthogonal unit base tensors in conjunction with two new sets of three invariants are employed for the representation of arbitrary isotropic tensor valued and scalar valued functions. The first set consists of the normalized identity tensor and the normalized deviatoric stress tensor (or deviatoric elastic strain tensor) along with a unit deviatoric tensor orthogonal to both of them. The second set consists of the normalized identity tensor and the other two unit deviatoric tensors which are coaxial with the stress tensor (or elastic strain tensor) and have the Lode angle of0and π/2respectively. Geometrically, the base tensors are characterized by three mutually orthogonal unit vectors and the three invariants are regarded as the components of a vector in principal space. With them, both the elastic constitutive equations and the flow rule of plasticity can be represented as a simple relationship among vectors in principal space.Two return mapping algorithms employing the representation are formulated in invariant space (principal space) and dimensions of the problem are reduced from six down to three. In contrast to the conventional approaches in principal space, the procedure of transformation between the principal stress space and the general stress space is omitted and the explicit computation of the principal axes is avoided. In addition, expressions involved and the matrix which needs to be inversed typically take the simple form.The expressions for the consistent tangent operator for the proposed algorithm are derived in an efficient and closed-form manner based on the two sets of mutually orthogonal unit basis tensors.The proposed methods are implemented into the commercial finite element software package ABAQUS, and the results and computational time of several numerical examples are compared between proposed methods and other available methods. It shows that the proposed methods are more efficient, and the method based on the first set base tensor is more efficient than the one based on the second base tensor.Finally, the proposed methods are extended to the finite deformation formulation and are also proved to be more efficient.